- spherical geometry -
There is some point and line so that there are no parallels.

Notice that number 1. holds in spherical geometry because any two "lines" (ie great circles) intersect at 2 points - at opposite points on the sphere. Hence I can't find any non-intersecting great circles on the sphere and so there are no parallels on the sphere (recall that non-equator latitudes are not shortest distance paths so they are not considered to be "lines".)

- hyperbolic geometry -
There is some point and line so that there are
**at least 2**parallels.We are going to work our way towards number 2.

**Next week we will see real-life applications of hyperbolic geometry.**

The following file is an interactive version of the model.
Drag the points H, G and I around to see what happens to the
sum of the angles in the resulting hyperbolic triangle.

Interactive
Poincare disk angle sum

**Question 1** How large can the sum of the angles of a hyperbolic
triangle
get?

**Question 2** What kind of triangle results in a large angle sum?
Explain why this makes sense by using the visualization of this model
(above) and an argument similar to the one we made to explain why
small spherical triangles have angle sums close to 180 degrees.

**Question 3**
How small can the sum of the angles of a hyperbolic triangle get?

The following picture shows the Poincare disk model with three points
X, Y and Z. I have measured angle XYZ and m[2] shows me that the measure
of this angle is about 90 degrees. Hence XYZ forms a right triangle
with XZ as the hypotenuse.
I then calculated XY^{2} + YZ^{2} and compared it to XZ
^{2} to see whether the
Pythagorean theorem holds in this model. We see that for this
triangle XY^{2} + YZ^{2} - XZ^{2}
= -.335 and so we see that
XY^{2} + YZ^{2} < XZ^{2} by .335.
Hence the Pythagorean theorem does not
hold for this triangle in this model.

The following file is an interactive version of the model.
Interactive Poincare Disk
Pythagorean theorem
**Question 1:**
Drag the points to make a small right triangle.
Be sure that the points don't
touch and be sure that the angle (m[2])
is about 90 degrees.
**Part a:** What is the measure of the angle?
**Part b:** What is
XY^{2} + YZ^{2} - XZ^{2}?
**Question 2:**
Drag the points to make a large right triangle and be sure that the angle
(m[2]) is about 90 degrees.
**Part a:** What is the measure of the angle?
**Part b:** What is
XY^{2} + YZ^{2} - XZ^{2}?
**Question 3:**
Given this model, why does it make sense that for small right
hyperbolic triangles the calculation
XY^{2} + YZ^{2} - XZ^{2}
is closer to 0 than is the same calculation for large right hyperbolic
triangles?